Shape preserving constrained and monotonic rational quintic fractal interpolation functions
Chand, A.; Tyada, K.
2018-06-02 00:00:00
Shape preserving interpolants play important role in applied science and engineering. In this paper, we develop a new class of
$${\mathscr {C}}^2$$
C
2
-rational quintic fractal interpolation function (RQFIF) by using rational quintic functions of the form
$$\frac{p_i(t)}{q_i(t)},$$
p
i
(
t
)
q
i
(
t
)
,
where
$$p_i(t)$$
p
i
(
t
)
is a quintic polynomial and
$$q_i(t)$$
q
i
(
t
)
is a cubic polynomial with two shape parameters. The convergent result of the RQFIF to a data generating function in
$${\mathscr {C}}^3$$
C
3
is presented. We derive simple restrictions on the scaling factors and shape parameters such that the developed rational quintic FIF lies above a straight line when the interpolation data with positive functional values satisfy the same constraint. Developing the relation between the attractors of equivalent dynamical systems, the constrained RQFIF can be extended to any general data. The positivity preserving RQFIF is a particular case of our result. In addition to this we also deduce the range on the IFS parameters to preserve the monotonicity aspect of given restricted type of monotonic data. The second derivative of the proposed RQFIF is irregular in a finite or dense subset of the interpolation interval, and matches with the second derivative of the classical rational quintic interpolation function whenever all scaling factors are zero. Thus, our scheme outperforms the corresponding classical counterpart, and the flexibility offered through the scaling factors is demonstrated through suitable examples.
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Abstract

Shape preserving interpolants play important role in applied science and engineering. In this paper, we develop a new class of
$${\mathscr {C}}^2$$
C
2
-rational quintic fractal interpolation function (RQFIF) by using rational quintic functions of the form
$$\frac{p_i(t)}{q_i(t)},$$
p
i
(
t
)
q
i
(
t
)
,
where
$$p_i(t)$$
p
i
(
t
)
is a quintic polynomial and
$$q_i(t)$$
q
i
(
t
)
is a cubic polynomial with two shape parameters. The convergent result of the RQFIF to a data generating function in
$${\mathscr {C}}^3$$
C
3
is presented. We derive simple restrictions on the scaling factors and shape parameters such that the developed rational quintic FIF lies above a straight line when the interpolation data with positive functional values satisfy the same constraint. Developing the relation between the attractors of equivalent dynamical systems, the constrained RQFIF can be extended to any general data. The positivity preserving RQFIF is a particular case of our result. In addition to this we also deduce the range on the IFS parameters to preserve the monotonicity aspect of given restricted type of monotonic data. The second derivative of the proposed RQFIF is irregular in a finite or dense subset of the interpolation interval, and matches with the second derivative of the classical rational quintic interpolation function whenever all scaling factors are zero. Thus, our scheme outperforms the corresponding classical counterpart, and the flexibility offered through the scaling factors is demonstrated through suitable examples.

Journal

International Journal of Advances in Engineering Sciences and Applied Mathematics
– Springer Journals